The unitary duals of the complex exceptional simple Lie groups $F_4$, $E_6$, $E_7$, and $E_8$

For a real reductive Lie group $G$, the unitary dual $\widehat{G}$ is the set of irreducible unitary representations of $G$, and its classification is a central open problem of representation theory. When a complex simple Lie group is viewed as a real Lie group, $\widehat{G}$ was determined long ago for $GL(n,\mathbb{C})$, for the other complex classical groups, and for complex $G_2$, but it remains unknown for the complex exceptional groups of type $F_4$, $E_6$, $E_7$, $E_8$. Vogan's fundamental parallelepiped (FPP) conjecture — proved for complex simple groups by Dong and Wong and for arbitrary real reductive groups by Davis and Mason-Brown — reduces the classification to a finite task: every irreducible unitary representation with real infinitesimal character is cohomologically induced in the weakly good range from unitary representations whose real infinitesimal character lies inside the FPP, a bounded polytope containing only finitely many infinitesimal characters. The extension of the FPP conjecture to all real reductive groups was resolved by Davis and Mason-Brown in 2024. The surviving open problem is to carry out the finite determination explicitly: enumerate, for each complex exceptional group, the irreducible $(\mathfrak{g},K)$-modules with infinitesimal character inside the FPP, decide which are unitary, and thereby complete the classification of the unitary duals of all complex simple Lie groups.

Background

Let $G$ be a real reductive Lie group, for instance a group in Harish-Chandra's class (a connected reductive group with a Cartan involution and compatible maximal compact subgroup $K$). The unitary dual $\widehat{G}$ is the set of equivalence classes of irreducible unitary representations of $G$ on Hilbert spaces. By Harish-Chandra theory these are studied algebraically as irreducible $(\mathfrak{g},K)$-modules — the $\mathfrak{g}$-modules of $K$-finite vectors, where $\mathfrak{g}$ is the complexified Lie algebra — carrying a positive-definite invariant Hermitian form. Within the unitary dual, the tempered representations — those whose matrix coefficients lie in $L^{2+\varepsilon}$ for every $\varepsilon>0$, roughly the representations contributing to the Plancherel decomposition of $L^2(G)$ — form the best-understood part. Classifying $\widehat{G}$ for all real reductive groups is one of the fundamental open problems of the field, going back to the origins of the subject.

A complex simple Lie group may itself be regarded as a real Lie group; its maximal compact subgroup $K$ is then a compact real form of $G$. Irreducible admissible $(\mathfrak{g},K)$-modules of such a group are parametrized by pairs of weights $(\lambda_L,\lambda_R)$ modulo the Weyl group, and their restrictions to $K$ decompose into irreducible representations of $K$ (K-types). Every irreducible module has an infinitesimal character: through the Harish-Chandra isomorphism, the center of the universal enveloping algebra acts via the Weyl-group orbit $\chi_\lambda$ of a functional $\lambda$ on a Cartan subalgebra, where the half-sum $\rho$ of the positive roots corresponds to the trivial representation. The infinitesimal character is called real when $\lambda$ lies in the real span of the weight lattice, and dominant when $\langle\lambda,\alpha^\vee\rangle\ge 0$ for all positive coroots $\alpha^\vee$ (the natural pairing of a functional with a coroot).

For complex groups viewed as real groups, the unitary dual was determined for $GL(n,\mathbb{C})$ by Vogan (1986), for the remaining complex classical groups by Barbasch (1989), and for complex $G_2$ by Vogan (1994). The complex exceptional groups of type $F_4$, $E_6$, $E_7$, $E_8$ have remained out of reach.

Two reduction principles organize the modern attack. First, Salamanca-Riba and Vogan (1998) conjectured that the classification reduces to representations containing unitarily small K-types — K-types whose highest weights lie in a certain bounded convex hull — so that only finitely many K-types, and hence essentially finitely many candidates, need to be examined. Second, in 2023 Vogan sharpened this to the fundamental parallelepiped (FPP) conjecture. The FPP is the region of dominant real functionals defined by $0\le\langle\lambda,\alpha^\vee\rangle\le 1$ for every simple coroot $\alpha^\vee$. The conjecture asserts that if an irreducible $(\mathfrak{g},K)$-module with real infinitesimal character $\chi_\lambda$ satisfies $\langle\lambda,\alpha^\vee\rangle> 1$ for some simple coroot, and it is not cohomologically induced from a proper Levi subgroup, then it is not unitary. Here cohomological induction is the derived-functor construction producing representations of $G$ from representations of the Levi factor of a $\theta$-stable parabolic subgroup; in the weakly good range (a positivity range for the inducing infinitesimal character) it preserves and reflects unitarity, as recorded by both Dong–Wong and Davis–Mason-Brown. The conjecture therefore implies that every irreducible unitary representation with real infinitesimal character is assembled, by unitarity-preserving cohomological induction, from unitary representations with real infinitesimal character inside the FPP — a finite set, because the FPP is a bounded polytope that meets the infinitesimal-character lattice in finitely many points, and above each such infinitesimal character there are only finitely many irreducibles.

Dong and Wong proved the FPP conjecture for all complex simple Lie groups (2024), and Wong proved it for $U(p,q)$ (2024). Davis and Mason-Brown (2024) then proved it for arbitrary real reductive groups, including non-linear coverings, by a case-free argument combining Beilinson–Bernstein localization, which realizes representations as twisted $\mathcal{D}$-modules (sheaves of modules over rings of differential operators) on the flag variety (the projective variety parametrizing the Borel subgroups of $G$), with the Hodge filtration of mixed Hodge modules (a sheaf-theoretic refinement of Hodge theory adapted to singular spaces), resting on the unitarity criterion of Davis and Vilonen. In consequence, the classification of the unitary dual of every complex simple group is now reduced to a finite determination. For the complex classical groups and $G_2$ the answer was already known; what remains is the four complex exceptional groups.

Substantial partial information exists for them. The Dirac series — the unitary representations with non-zero Dirac cohomology, where Dirac cohomology is the kernel on K-types of a Dirac operator built from the Clifford algebra of $\mathfrak{g}/\mathfrak{k}$ — have been classified for all complex simple groups, complex $E_8$ being completed by Barbasch and Wong (2026), who replaced memory-exhausting computations in the atlas software (the computational platform of the Atlas of Lie Groups and Representations project) by structural arguments with bottom-layer K-types (a distinguished class of small K-types that are forced to occur in cohomologically induced modules) and intertwining operators (linear maps that commute with the group action between two representations). However, the Dirac series is in general a proper subset of the unitary dual: continuous complementary-series families — one-parameter families of unitary representations interpolating between tempered endpoint representations — typically have zero Dirac cohomology. The conjectured building blocks of the dual — the unipotent representations attached to nilpotent coadjoint orbits (orbits of the group acting on the nilpotent elements of the dual of its Lie algebra) — are also not yet shown to account for everything unitary. The full unitary duals of the complex groups of type $F_4$, $E_6$, $E_7$, $E_8$ are therefore still undetermined.

Problem Statement

Determine the unitary duals of the complex exceptional simple Lie groups of type $F_4$, $E_6$, $E_7$, and $E_8$, each viewed as a real Lie group. The answer can depend on the isogeny class (the choice of group among those related by finite covering homomorphisms — for instance, the simply connected and adjoint forms of $E_6$ and $E_7$ should be distinguished), and every isogeny class of each type must be treated, with each answer specifying the isogeny class it describes.

By the FPP theorem — proved for complex simple groups by Dong and Wong and for arbitrary real reductive groups by Davis and Mason-Brown — every irreducible unitary $(\mathfrak{g},K)$-module of such a group with real infinitesimal character is cohomologically induced in the weakly good range from irreducible unitary modules whose real dominant infinitesimal character lies inside the fundamental parallelepiped $\{\lambda:\ 0\le\langle\lambda,\alpha^\vee\rangle\le 1\ \text{for every simple coroot }\alpha^\vee\},$ and cohomological induction in the weakly good range preserves unitarity in both directions. This statement governs representations with real infinitesimal character; the non-real-infinitesimal-character part of the dual is reached by the classical reduction to real infinitesimal character, according to which every irreducible unitary $(\mathfrak{g},K)$-module is a unitary parabolic induction $\operatorname{Ind}_P^G(\pi_{\mathrm{re}}\otimes\chi)$ of the tensor product of a unitary module $\pi_{\mathrm{re}}$ with real infinitesimal character and a unitary character $\chi$ of the Levi factor, so that the non-real-infinitesimal-character unitary modules are assembled from the real-infinitesimal-character unitary modules by unitary parabolic induction. The classification therefore reduces to a finite determination, and the problem is to carry it out explicitly for each of the four groups:

1. enumerate the finite set of irreducible $(\mathfrak{g},K)$-modules whose infinitesimal character lies inside the FPP;
2. decide, with proof, which of these modules are unitary and which are not;
3. assemble from these data the complete list of irreducible unitary representations of the group, recording the cohomological-induction data by which the real-infinitesimal-character part of $\widehat{G}$ and the unitary parabolic inductions by which its non-real-infinitesimal-character part are recovered from the FPP-finite unitary set.

A complete determination of $\widehat{G}$ for each complex exceptional group consists of a unitary/non-unitary verdict for every irreducible module with infinitesimal character inside the FPP, together with the cohomological-induction data assembling the real-infinitesimal-character part of the dual from the FPP-finite unitary set and the classification of the unitary representations with non-real infinitesimal character; it must agree with the previously known sub-cases, the tempered dual and the already-classified Dirac series.

References

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